Saving and Borrowing


BUSI 721
Jones Graduate School of Business
Rice University

Kerry Back

Saving

  • Funds can be kept as “cash,” earning money market rate.
  • \(n\) risky assets, \(w_i\) = weights, \(x_s \ge 0\) = fraction saved

\[x_s + \sum_{i=1}^n w_i = 1\]

  • \(r_s\) = money market rate (saving rate)
  • portfolio expected return is \(x_sr_s + w^\top \bar{r}\)
  • portfolio variance is still \(w^\top C w\)

Single risky asset

  • \(w\) = risky asset allocation, \(\bar{r}\) = mean, \(\sigma\) = std dev
  • \(x_s = 1-w\)
  • portfolio expected return is

\[ (1-w)r_s + w \bar{r} = r_s + (\bar{r}-r_s)x_s\]

  • portfolio standard deviation is \(\sigma w\)

Mean – Standard Deviation Plot

  • \(\bar{r}\) = 10%, \(\sigma\) = 15%, \(r_s\) = 2%
  • slope is Sharpe ratio \((\bar{r}-r_s)/\sigma\)

Borrowing (Margin loans)

  • \(x_b \ge 0\) is fraction borrowed, \(r_b\) = borrowing rate
  • portfolio expected return is \(w^\top \bar{r} - x_br_b\)
  • portfolio variance is still \(w^\top C w\)
  • with single risky asset, \(x_b = w-1\)
    • portfolio expected return is

\[w \bar{r} - (w-1)r_b = r_b + (\bar{r}-r_b)w\]

  • portfolio standard deviation is \(\sigma w\)

Mean – Standard Deviation Plot

  • \(\bar{r}\) = 10%, \(\sigma\) = 15%, \(r_b\) = 5%
  • slope is Sharpe ratio \((\bar{r}-r_b)/\sigma\)

Saving and borrowing opportunities

  • \(\bar{r}\) = 10%, \(\sigma\) = 15%, \(r_s\) = 2%, \(r_b\) = 5%
  • slopes are Sharpe ratios \((\bar{r}-r_s)/\sigma\) and \((\bar{r}-r_b)/\sigma\)